Raven paradox: problem and solution given on the basis of Aristotle’s logic

The paper is devoted to the solution of two well-known paradoxes of inductive logic: Hempel’s and Goodman’s, which the science has not solved unambiguously yet. The results of this research can be used in any natural science, but they are especially relevant for areas where there is an emphasis placed on environmental friendliness and sustainable development. The central problem of this research is the problem of limits of applying classical logic laws. The problem is solved by method of reduction of logical laws to those cases where they, according to Aristotle, act faultlessly, and refusal of their recognition in the cases where their action is questionable. The aim of the paper is to demonstrate the solution of both problems within Aristotle’s logic. In that regard, the following results are received: common faults of previous solutions of Hempel’s paradox, consisting in ignoring any of its parties, are revealed; the original nature of Goodman’s paradox, consisting in wrong interpreting “inductive confirmation” criteria is opened; two methods of forming and assessing the subject volumes of statements are revealed: analytical and synthetical ones; it is proved that the theses treated in Hempel’s paradox as equivalent ones are not always so, but only on condition of their subjects’ reality and of their subject volumes’ identity; it is established that the conditions of the statement equivalence correspond to the limits of applying logic laws in Aristotle’s interpretation.

Hempel’s Raven Revisited

The paper takes a novel approach to a classic problem—Hempel’s Raven Paradox. A standard approach to it supposes the solution to consist in bringing our inductive logic into “reflective equilibrium” with our intuitive judgements about which inductive inferences we should license. This approach leaves the intuitions as a kind of black box and takes it on faith that, whatever the structure of the intuitions inside that box might be, it is one for which we can construct an isomorphic formal edifice, a system of inductive logic. By popping open the box we can see whether that faith is misplaced. I aim, therefore, to characterize our pre-theoretical, intuitive understanding of generalizations like “ravens are black” and argue that, intuitively, we take them to mean, for instance: “ravens are black by some indeterminate yet characteristic means.” I motivate and explicate this formulation and bring it to bear on Hempel’s Problem.

Channels’ Confirmation and Predictions’ Confirmation: From the Medical Test to the Raven Paradox

After long arguments between positivism and falsificationism, the verification of universal hypotheses was replaced with the confirmation of uncertain major premises. Unfortunately, Hemple proposed the Raven Paradox. Then, Carnap used the increment of logical probability as the confirmation measure. So far, many confirmation measures have been proposed. Measure F proposed by Kemeny and Oppenheim among them possesses symmetries and asymmetries proposed by Elles and Fitelson, monotonicity proposed by Greco et al., and normalizing property suggested by many researchers. Based on the semantic information theory, a measure b* similar to F is derived from the medical test. Like the likelihood ratio, measures b* and F can only indicate the quality of channels or the testing means instead of the quality of probability predictions. Furthermore, it is still not easy to use b*, F, or another measure to clarify the Raven Paradox. For this reason, measure c* similar to the correct rate is derived. Measure c* supports the Nicod Criterion and undermines the Equivalence Condition, and hence, can be used to eliminate the Raven Paradox. An example indicates that measures F and b* are helpful for diagnosing the infection of Novel Coronavirus, whereas most popular confirmation measures are not. Another example reveals that all popular confirmation measures cannot be used to explain that a black raven can confirm “Ravens are black” more strongly than a piece of chalk. Measures F, b*, and c* indicate that the existence of fewer counterexamples is more important than more positive examples’ existence, and hence, are compatible with Popper’s falsification thought.

A MORE DEVASTATING VERSION OF THE RAVEN PARADOX

Hempel's famous Raven Paradox derives from Nicod's criteria for confirmation and the Equivalence Condition, the unintuitive conclusion that things like white roses, green T-shirts and ice cubes confirm the raven hypothesis ‘All ravens are black.’ By a small rearrangement of the Equivalence Condition, I show that we can also derive the conclusion, which sounds even more intuitively intolerable, that observation of black ravens fails to confirm the raven hypothesis. We are left with the contradictory result that black ravens both confirm and do not confirm the raven hypothesis.